Abstract
The main aim of this paper is to study whether the Gromov hyperbolicity is preserved under some transformations on Riemann surfaces (with their Poincaré metrics). We prove that quasiconformal maps between Riemann surfaces preserve hyperbolicity; however, we also show that arbitrary twists along simple closed geodesics do not preserve it, in general.
Similar content being viewed by others
References
Gromov, M.: Hyperbolic Groups, Essays in Group Theory, Math. Sci. Res. Inst. Publ., 8, Springer, New York, 1987, 75–263
Bonk, M., Schramm, O.: Embeddings of Gromov hyperbolic spaces. Geom. Funct. Anal., 10(2), 266–306 (2000)
Foertsch, T., Schroeder, V.: A product construction for hyperbolic metric spaces. Illinois J. Math., 49(3), 793–810 (2005)
Naor, A., Peres, Y., Schramm, O., et al.: Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces. Duke Math. J., 134(1), 165–197 (2006)
Wenger, S.: Gromov hyperbolic spaces and the sharp isoperimetric constant. Invent. Math., 171(1), 227–255 (2008)
Balogh, Z. M., Buckley, S. M.: Geometric characterizations of Gromov hyperbolicity. Invent. Math., 153, 261–301 (2003)
Benoist, Y.: Convexes hyperboliques et fonctions quasisymétriques. Publ. Math. Inst. Hautes Études Sci., 97, 181–237 (2003)
Bonk, M., Heinonen, J., Koskela, P.: Uniformizing Gromov hyperbolic spaces, Astérisque 270, 2001
Breda, A. M., Sigarreta, J. M.: Symmetry and transitive properties of monohedral f-triangulations of the Riemannian sphere. Acta Mathematica Sinica, English Series, 25, 1609–1616 (2009)
Hästö, P. A.: Gromov hyperbolicity of the j G and \( \tilde jG \) metrics. Proc. Amer. Math. Soc., 134, 1137–1142 (2006)
Hästö, P. A., Lindén, H., Portilla, A., et al.: Gromov hyperbolicity of Denjoy domains with hyperbolic and quasihyperbolic metrics. J. Math. Soc. Japan, to appear
Hästö, P. A., Portilla, A., Rodríguez, J. M., et al.: Gromov hyperbolic equivalence of the hyperbolic and quasihyperbolic metrics in Denjoy domains. Bull. London Math. Soc., 42(2), 282–294 (2010)
Hästö, P. A., Portilla, A., Rodríguez, J. M., et al.: Uniformly separated sets and Gromov hyperbolicity of domains with the quasihyperbolicity metric. Medit. J. Math., DOI: 10.1007/s00009-010-0059-7
Hästö, P. A., Portilla, A., Rodríguez, J. M., et al.: Comparative Gromov hyperbolicity results for the hyperbolic and quasiyperbolic metrics. Complex Var. Elliptic Equ., 55, 127–135 (2010)
Karlsson, A., Noskov, G. A.: The Hilbert metric and Gromov hyperbolicity. Enseign. Math., 48, 73–89 (2002)
Lindén, H.: Gromov hyperbolicity of certain conformal invariant metrics. Ann. Acad. Sci. Fenn. Math. 32(1), 279–288 (2007)
Sun, M. F., Shen, Y. L.: On holomorphic sections in Teichmuller spaces. Acta Math. Sinica, English Series, 25, 2023–2034 (2009)
Alvarez, V., Portilla, A., Rodríguez, J. M., et al.: Gromov hyperbolicity of Denjoy domains. Geom. Dedicata, 121, 221–245 (2006)
Portilla, A., Rodríguez, J. M., Tourís, E.: Gromov hyperbolicity through decomposition of metric spaces II. J. Geom. Anal., 14, 123–149 (2004)
Portilla, A., Rodríguez, J. M., Tourís, E.: The topology of balls and Gromov hyperbolicity of Riemann surfaces. Diff. Geom. Appl., 21, 317–335 (2004)
Portilla, A., Rodríguez, J. M., Tourís E.: The role of funnels and punctures in the Gromov hyperbolicity of Riemann surfaces. Proc. Edinb. Math. Soc., 49, 399–425 (2006)
Portilla, A., Rodríguez, J. M., Tourís, E.: A real variable characterization of Gromov hyperbolicity of flute surfaces. Osaka J. Math., to appear
Portilla, A., Rodríguez, J. M., Tourís, E.: Structure Theorem for Riemannian surfaces with arbitrary curvature. Preprint
Portilla, A., Tourís, E.: A characterization of Gromov hyperbolicity of surfaces with variable negative curvature. Publ. Mat., 53, 83–110 (2009)
Rodríguez, J. M., Tourís, E.: Gromov hyperbolicity through decomposition of metric spaces. Acta Math. Hung., 103, 53–84 (2004)
Rodríguez, J. M., Tourís, E.: A new characterization of Gromov hyperbolicity for Riemann surfaces. Publ. Mat., 50, 249–278 (2006)
Rodríguez, J. M., Tourís, E.: Gromov hyperbolicity of Riemann surfaces. Acta Mathematica Sinica, English Series, 23, 209–228 (2007)
Tourís, E.: Graphs and Gromov hyperbolicity of non-constant negatively curved surfaces. J. Math. Anal. Appl., to appear
Ghys, E., de la Harpe, P.: Sur les Groupes Hyperboliques d’après Mikhael Gromov, Progress in Mathematics, Volume 83, Birkhäuser, 1990
Basmajian, A.: Hyperbolic structures for surfaces of infinite type. Trans. Amer. Math. Soc., 336, 421–444 (1993)
Haas, A.: Dirichlet points, Garnett points and infinite ends of hyperbolic surfaces I. Ann. Acad. Sci. Fenn. Series AI, 21, 3–29 (1996)
Haas, A., Susskind, P.: The geometry at infinity of a hyperbolic Riemann surface of infinite type. Geom. Dedicata, 130, 1–24 (2007)
Buser, P.: Geometry and Spectra of Compact Riemann Surfaces, Birkhäuser, Boston, 1992
Chavel, I.: Eigenvalues in Riemannian Geometry, Academic Press, New York, 1984
Alvarez, V., Rodríguez, J. M.: Structure theorems for Riemann and topological surfaces. J. London Math. Soc., 69, 153–168 (2004)
Douady, A., Earle, C.: Conformally natural extension of homeomorphisms of the circle. Acta Math., 157, 23–48 (1986)
Matsuzaki, K.: Quasiconformal mapping class groups having common fixed points on the asymptotic Teichm üller spaces. J. d’Analyse Math., 102, 1–28 (2007)
Anderson, J. W.: Hyperbolic Geometry, Springer, London, 1999
Bers, L.: An Inequality for Riemann Surfaces. Differential Geometry and Complex Analysis, H. E. Rauch Memorial Volume, Springer-Verlag, 1985
Beardon, A. F.: The geometry of discrete groups, Springer-Verlag, New York, 1983
Matsuzaki, K.: A countable Teichmüller modular group. Trans. Amer. Math. Soc., 357, 3119–3131 (2005)
Author information
Authors and Affiliations
Corresponding author
Additional information
The second author is supported in part by two grants from Ministerio de Ciencia e Innovación (MTM 2009-07800 and MTM 2008-02829-E), Spain
Rights and permissions
About this article
Cite this article
Matsuzaki, K., Rodríguez, J.M. Twists and Gromov hyperbolicity of riemann surfaces. Acta. Math. Sin.-English Ser. 27, 29–44 (2011). https://doi.org/10.1007/s10114-011-9693-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-011-9693-7